SOLUTION: The sides of a triangle are k, k - 3, and k + 3. If the area of the triangle is √1200 find the value of k.

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Question 256378: The sides of a triangle are k, k - 3, and k + 3. If the area of the triangle is
√1200 find the value of k.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

Heron's formula for the area A of a triangle with
sides a,b, and c is

A=sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29 where s=%28a%2Bb%2Bc%29%2F2

We will eventually let a=k, b=k+-+3, c=k+%2B+3,
and A=sqrt%281200%29

But to make things easier we will first simplify
Heron's formula by first removing the square root and
secondly, eliminating s:

A=sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29

A%5E2=s%28s-a%29%28s-b%29%28s-c%29

where





so, substituting to eliminate s,

A%5E2=s%28s-a%29%28s-b%29%28s-c%29



A%5E2=%28%28a%2Bb%2Bc%29%28-a%2Bb%2Bc%29%28a-b%2Bc%29%28a%2Bb-c%29%29%2F16

16A%5E2=%28a%2Bb%2Bc%29%28-a%2Bb%2Bc%29%28a-b%2Bc%29%28a%2Bb-c%29

Now that we have rewritten Heron's formula in terms of the
sides only, we simplify the factors to substitute in
the above revised version of Heron's formula, which I think
is superior to the original version, especially for 
calculating purposes.

So now we substitute a=k, b=k+-+3, c=k+%2B+3, and A=sqrt%281200%29

A%5E2=%28sqrt%281200%29%29%5E2=1200
a%2Bb%2Bc+=+k%2B%28k-3%29%2B%28k%2B3%29=k%2Bk-3%2Bk%2B3=3k
-a%2Bb%2Bc+=+-k%2B%28k-3%29%2B%28k%2B3%29=-k%2Bk-3%2Bk%2B3=k
a-b%2Bc+=+k-%28k-3%29%2B%28k%2B3%29=k-k%2B3%2Bk%2B3=k%2B6
a%2Bb-c+=+k%2B%28k-3%29-%28k%2B3%29=k%2Bk-3-k-3=k-6

Substituting these factors:

16A%5E2=%28a%2Bb%2Bc%29%28a%2Bb-c%29%28a-b%2Bc%29%28-a%2Bb%2Bc%29%29

16%281200%29=%283k%29%28k-6%29%28k%2B6%29%28k%29

19200=3k%5E2%28k-6%29%28k%2B6%29

Divide both sides by 3

6400=k%5E2%28k-6%29%28k%2B6%29

6400=k%5E2%28k%5E2-36%29

6400=k%5E4-36k%5E2

0=k%5E4-36k%5E2-6400

0=%28k%5E2-100%29%28k%5E2%2B64%29

%28k%5E2-100%29%28k%5E2%2B64%29=0

%28k-10%29%28k%2B10%29%28k%5E2%2B64%29=0

The only real positive solution is
gotten by setting the first factor = 0

k-10=0
k=10

Edwin